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The Husimi $Q$ function of a quantum state $\rho $ is defined as $ Q (\alpha)=\langle \alpha \vert \rho \vert \alpha \rangle $, where $\alpha = (x, p) $ is a phase space coordinate and $\vert \alpha \rangle$ is a coherent state.

Is the off-diagonal generalization $ Q (\alpha, \beta)=\langle \alpha \vert \rho \vert \beta \rangle $ used for anything? Does it have a name?

This is an interesting object because it essentially measures coherence (or decoherence) in the overcomplete basis of wavepackets .

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I finally found some prior art. This object has been introduced as the "Husimi Matrix" by Harriman

"Some properties of the Husimi function" Harriman, John E., The Journal of Chemical Physics, 88, 6399-6408 (1988),
http://dx.doi.org/10.1063/1.454477

and briefly referred to by Morrison and Parr

"Approximate density matrices and Husimi functions using the maximum entropy formulation with constraints"
Morrison, Robert C. and Parr, Robert G., International Journal of Quantum Chemistry, 39: 823–837 http://dx.doi.org/10.1002/qua.560390607

The treatment was fairly basic. From what I can tell, Harriman primarily introduced the Husimi matrix to highlight an analogy with density matrices (since you can use it as the kernel of an integral operator). Morrison and Parr use it for something related to calculating a density matrix as a maximum entropy Husimi function, but I don't really understand.

I do not believe anyone has explored the relationship to decoherence.

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